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```_______________________________________________________________________
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Therefore,
n C 2 + n C 5 + n C 8 + ...
=
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=
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=
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W.35 Ex.B Q.2, 4, 5
H.W. : Ex.B Q.1, 7, 9
Ex.C Q.2
Primitive Root
An n th root of unity z is said to be primitive if there doesn't exist a positive integer m less than n for
which z m = 1 .
z9 = 1
e.g.
Consider z3 =
2ï°
i
For z3 = e 3
â´
ï
2ï°
i
3
zk =
e
,
i
e
2kï°
9
and z 5 =
z33 =
ei2 ï°
for k = 0, 1, 2, ... , 8
10 ï°
i
9 .
e
= 1.
There exists a positive integer n less than 9 such that z
Let n be a positive integer such that z5n = 1 ï
ï
ï¦ 10ï° ï¶
ï¦ 10ï° ï¶
cos ï§
ï· n + i sin ï§
ï· n=1
ï¨ 9 ï¸
ï¨ 9 ï¸
ï
ï¦ 10ï° ï¶
cos ï§
ï· n =1
ï¨ 9 ï¸
ï
ï¦ 10ï° ï¶
ï§
ï· n = 2kï°
ï¨ 9 ï¸
ï
5n
= k
9
â´
9 divides n.
â´
9<n
and
e
n
3
ï¦ 10ï° ï¶
iï§
ï·n
ï¨ 9 ï¸
=1 .
=1
ï¦ 10ï° ï¶
sin ï§
ï·n = 0
ï¨ 9 ï¸
for some integer k
ï z5 is a primitive root of unity.
Theorem :
e
i
2kï°
n is a primitive n th root of 1 if k, n are relatively prime to each other .
P.M./Complex/p.21
```
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